684 Lord Bayleigh on Hamilton's Principle and 
be the curvatures of the images, as formed by rays in the 
two planes, 
l/p 1 = 2C, l//o 3 = 2D. . . . (24) 
The condition of astigmatism is then 
C = D; (25) 
but unless both constants vanish the image is curved. 
Finally the term containing E represents distortion. 
If we impose no restriction upon the values of the coustants 
of aberration, we have in general from (21), (22) 
dx/dl = A(3l 2 -f m 2 ) + 6B*7 + O' 2 , 
dyjdm = A {I 2 + 3m 2 ) + 2B x'l + V>x' 2 . 
These equations may be applied to find the curvatures of the 
image as formed by rays infinitely close to given rays, as for 
example when the aperture is limited by a narrow stop placed 
centrally on the axis, but otherwise arbitrarily. The principal 
ray is then characterized by the condition m = 0, and 
we have 
dx/dl = 3AZ 2 + 6Ba'Z + (V 2 = 3H + K, . . . (26) 
dy/dm=Al 2 + 2~Bx'l + Dx /2 = H + K, . . . (27) 
equations which determine the curvatures of the images as 
formed by rays in the neighbourhood of the given one, 
and deviating from it in the primary and secondary planes 
respectively. 
According to (26), (27), 
2B. =2Al 2 + 4:Bx / l + (C-B)x /2 } . . . (28) 
2K=(3D-0>' 2 (29) 
The requirement of flatness in both images is thus satisfied 
if H = 0, K = 0. The former is the condition of astigmatism, 
and it involves the ratio of x'\ I, which is dependent upon the 
position of the stop ; but the latter does not depend on this 
ratio. It corresponds to the condition formulated by 
Coddington and later by Petzval. From {28), (29) we 
may of course fall back upon the conditions already laid 
down for the case where A=0, B = 0. 
The further pursuit of this subject requires a more parti- 
cular examination of what occurs when light is refracted at 
spherical surfaces. Keference may be made to Schwarz- 
schild *, who uses Hamilton's methods as applied to a special 
form of the characteristic function designated as Seidel's 
Eikonal. A concise derivation of the Coddington-Petzval 
* Gottingen Abh. iv. 1905. 
